Tunneling splittings of vibrationally excited states using general instanton paths

Abstract

A multidimensional semiclassical method for calculating tunneling splittings in vibrationally excited states of molecules using Cartesian coordinates is developed. It is an extension of the theory by Mil'nikov and Nakamura [J. Chem. Phys. 122, 124311 (2005)] to asymmetric paths that are necessary for calculating tunneling splitting patterns in multi-well systems, such as water clusters. Additionally, new terms are introduced in the description of the semiclassical wavefunction that drastically improves the splitting estimates for certain systems. The method is based on the instanton theory and builds the semiclassical wavefunction of the vibrationally excited states from the ground-state instanton wavefunction along the minimum action path and its harmonic neighborhood. The splittings of excited states are thus obtained at a negligible added numerical effort. The cost is concentrated, as for the ground-state splittings, in the instanton path optimization and the hessian evaluation along the path. The method can thus be applied without modification to many mid-sized molecules in full dimensionality and in combination with on-the-fly evaluation of electronic potentials. The tests were performed on several model potentials and on the water dimer.

Vibrational tunneling spectra of molecules with asymmetric wells: a combined vibrational configuration interaction and instanton approach

“…It was shown that this approach gives equivalent results to the standard WKB method in 1D 44 . Moreover, the ground-state TS obtained from Herring formula using the ground-state JFI wavefunctions 42 is identical to the standard instanton result derived from the steepest descent approximation of the partition function in the path integral formulation 31 .…”

“…( 3), inside the barrier, we employ the JFI wavefunctions instead, which we recently derived in Ref. 42. These are based on the WKB method in which the energy is treated as a term of order h1 and is moved to the transport equation, leaving the Hamilton-Jacobi equation energy independent.…”

“…For this purpose, vibrational energies are computed either from a 2 × 2 matrix involving corresponding states in the two wells, an 8 × 8 matrix involving 4 local states at both sides of the barrier, and a 16 × 16 matrix model. Thereby, we again calculate the local single-well states using VSCF/VCI, while the TM matrix elements are computed using the recently developed JFI method 42 .…”

“…( 9) has been neglected. It is known that, unless these terms exclusively contribute to the TM elements, they introduce the dependence of the TM element h νν ′ on the connection point 42 . Therefore, we omit them from the treatment here to separate the effect of U terms from the effect of asymmetry in the dependence of the TM element with the variation of the connection point S cp .…”

“…The exact quantal calculation of these elements requires a large basis set that can accurately represent the form of the wavefunction inside the barrier. Instead, we obtain them by means of Herring formula 45 in combination with the semiclassical wavefunctions from the instanton theory 42,43 . Since the only effect of matrix (1) is to mix local wavefunctions of different minima via tunneling, we refer to it as the tunneling matrix 29 .…”